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This question already has an answer here:

If you want to say "he bought," would you say:

Il a acheté la maison.

or

Il est acheté la maison.

because the first one would sound awkward. (Two "ah" sounds in a row.) I've heard the second one used like "Il est allé." So would I use "est" or "a" here?

marked as duplicate by Laure, Toto, Un francophone, Luke Sawczak, Maroon Oct 10 at 15:36

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The rules for when to use which auxiliary can actually be summed up pretty quickly by a flowchart.

This covers almost all cases and is a standard way of analyzing it:

flowchart to choose être or avoir

According to this flowchart, you would note that « acheter » is not used reflexively here, and it's not a Dr. Mrs. Vandertramp verb, so use « avoir ».

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